> Curvature invariants are scalar quantities constructed from tensors that represent curvature. One of the most basic polynomial curvature invariants in general relativity is the Kretschmann scalar. This study is an investigation of this curvature invariant and the connection of geometry to entropy and information of different metrics and black holes. The scalar gives the curvature of the spacetime as a function of the radial distance r in the vicinity as well as inside of the black hole. We derive the Kretschmann Scalar (KS) first for a fifth force metric that incorporates a Yukawa correction, then for a Yukawa type of Schwarzschild black hole, for a Reissner-Nordstrom black hole and finally an internal star metric. Then we investigate the relation and derive the curvature?s dependence on the entropy S and number of information N. Finally we discuss the settings in which the entropy?s full range of positive and negative values would have a meaningful interpretation. The Kretschmann scalar helps us understand the black hole?s appearance as a ?whole entity?. It can be applied in solar mass size black holes, neutron stars or supermassive black holes at the center of various galaxies.
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